Advertisements
Advertisements
Question
Show that 1 + i10 + i20 + i30 is a real number.
Solution
1 + i10 + i20 + i30
= 1 + (i4)2 .i2 + (i4)5 + (i4)7 .i2
= 1 + (1)2 (– 1 ) + (1)5 + (1)7 (– 1) ...[∵ i4 = 1, i2 = –1]
= 1 – 1 + 1 –1
= 0, which is a real number.
APPEARS IN
RELATED QUESTIONS
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
5i
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
Answer the following:
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Evaluate: i131 + i49
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
What is the reciprocal of `3 + sqrt(7)i`.
Multiplicative inverse of 1 + i is ______.
Which of the following is correct for any two complex numbers z1 and z2?
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a + ib.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Evaluate the following:
i35
